What does “not for nothing” mean?

Contra the other two answers here, “not for nothing, but” is indeed used, as Urban Dictionary defines it, as a hedge. It is in fact a verbal tic of Aaron Sorkin’s that drew attention through its overuse on The West Wing: Inside Aaron Sorkin’s Brain, Sorkinisms II: Not for Nothing. Most famously in the dialogue between Josh and Joe Quincy, where Quincy says “Not for nothing, but those you are trying to reach won’t understand that joke, and those who do you already have.” (And I still have trouble understanding what Sorkin meant by that.)

Not For Nothing: After two huff and puff answers about how this is not proper English, this definition is offered:

If I recall correctly, “not for nothing” is used when you’re about to say something
that the other person will probably disagree with. Typically, you’re offering advice
that’s prefaced with “Not for nothing..but etc, etc.”

As another poster commented, after Josh/Quincy was cited:

Yes I saw that too. Funny because Leo said “Not for nothing…” on last week’s late night West Wing re-run. I guess they’re going to have all the characters say it sooner or later.) Could it mean maybe: I’m not saying this with any ulterior motive (as in “not for anything), but.. Or I have nothing to gain in saying this but…
That’s the sense I have, but maybe not. And I didn’t understand Josh’s joke either.

See also What does “not for nothing” mean?, which differentiates the “not for nothing, but” Sorkin idiom from Jack London’s earlier usage (“Not for nothing had he been exposed to the pitiless struggles for life”), which other answerers seem to be presupposing.